Theorem int0 4966

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4512	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3478	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4957	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 230	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 263	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2729	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ∅c0 4322  ∩ cint 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-dif 3951  df-nul 4323  df-int 4951

This theorem is referenced by:  unissint  4976  uniintsn  4991  rint0  4994  intex  5337  intnex  5338  oev2  8522  fiint  9323  elfi2  9408  fi0  9414  cardmin2  9993  00lsp  20591  cmpfi  22911  ptbasfi  23084  fbssint  23341  fclscmp  23533  zarcmplem  32856  rankeq1o  35138  bj-0int  35977  heibor1lem  36672  ipoglb0  47609  mreclat  47612